How should one interpret the results when Bonferroni corrected p = 0.16 and the corresponding Cohen’s d = 0.56?
Although p=0.16 means non-significant, Bonferroni correction is sometimes considered as too conservative and producing false negative results especially for testing many hypotheses (for details see Blakesley et al. 2009 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3045855/).
Cohen's d of 0.56 is an effect size considered to be medium (calculated as a difference in means divided by standard deviation). For interpretation see http://rpsychologist.com/d3/cohend/.
Please do not confuse effect size with test significance (Sullivan and Feinn 2012 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3444174/)
It means just what it looks like --- a "medium"-sized observed effect which is not statistically significant at the alpha=.05 level.
As pointed out above, you have a medium sized effect that is not statistically significant. Depending on your sample size, that effect size may/may not be stable. It is possible that due to sampling error your effect size is overestimated, but that is contingent upon other factors of your data and design.
Thank you every one.
Michal thanks a lot for sharing for a very good article.
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